Sharp invariant relationships involving various types of domination numbers are found between a graph and its line graph.

A well-known problem in domination theory is the long-standing conjecture of V.G. Vizing from 1963 (see [7]) that the domination number of the Cartesian product of two graphs is at least as large as the product of the domination numbers of the individual graphs.
Although limited progress has been made, this problem essentially remains open. The usefulness of a maximum 2-packing in one of the graphs in establishing a lower bound has been recognized for some time.
In this paper, we shall extend this approach so as to take advantage of 2-packings whose membership can be altered in a certain way. This results in an improved lower bound for graphs which have 2-packings of this type.

A graph \(G\) is claw-free if it does not contain any complete bipartite graph \(K_{1,3}\) as an induced subgraph, and closed claw-free if it is the line-graph of a triangle-free graph. The inflation \(H_1\) of a graph \(H\) is obtained from \(\mathop{H}\limits^{i}\) by replacing each vertex \(x\) of degree \(d(x)\) by a clique \(X \simeq K_{d(x)}\).
Every inflated graph \(G = H_1\) is closed claw-free.
The minimum cardinalities \(\gamma(G)\), \(\text{ir}(G)\), and \(\text{rai}(G)\) of respectively a dominating set, a maximal irredundant set, and an \(R\)-annihilated irredundant set of any graph \(G\) satisfy
\(\text{rai}(G) \leq \text{ir}(G) \leq \gamma(G).\)
The motivation of this paper is that for inflated graphs, it is known that the difference \(\gamma(G) – \text{ir}(G)\) can be arbitrarily large, but not how large the ratio \(\gamma(G)/\text{ir}(G)\) can be. We show that \(\gamma(G) \leq 3\text{rai}(G)/2\) for every claw-free graph \(G\) and study the sharpness of the bounds
\(1 \leq {\gamma(G)}/{\text{ir}(G)} \leq {\gamma(G)}/{\text{rai}(G)} \leq {3}/{2}\)
in the four classes of claw-free graphs, closed claw-free graphs, inflated graphs, and line graphs of bipartite graphs.

Let \(\tau(G)\) denote the number of vertices in a longest path of the graph \(G = (V, E)\). A subset \(K\) of \(V\) is called a \(P_n\)-\({kernel}\) of \(G\) if \(\tau(G[K]) \leq n – 1\) and every vertex \(v \in V(G – K)\) is adjacent to an end-vertex of a path of order \(n – 1\) in \(G[K]\).
A partition \(\{A, B\}\) of \(V\) is called an \((a, b)\)-partition if \(\tau(G[A]) \leq a\) and \(\tau(G[B]) \leq b\).
We show that any graph with girth greater than \(n – 3\) has a \(P_n\)-kernel and that every graph has a \(P_\gamma\)-kernel. As corollaries of these results, we show that if \(\tau(G) = a + b\) and \(G\) has girth greater than \(a – 2\) or \(a \leq 6\), then \(G\) has an \((a, b)\)-partition.

In this paper, we establish that for arbitrary positive integers k and m, where \(k > 1\), there exists a tournament which has exactly m minimum dominating sets of order \(k\). A construction of such tournaments will be given.

A connected graph \(G\) is \((\gamma, k)\)-insensitive if the domination number \(\gamma(G)\) is unchanged when an arbitrary set of \(k\) edges is removed. The problem of finding the least number of edges in any such graph has been solved for \(k = 1\) and for \(k = \gamma(G) = 2\). Asymptotic results as \(n\) approaches infinity are known for \(k \geq 2\) and \(k+1 \leq \gamma(G) \leq 2k\). Note that for \(k = 2\), this bound holds only for graphs \(G$ with \(\gamma(G) \in \{3,4\}\). In this paper, we present an asymptotic bound for the minimum number of edges in an extremal \((\gamma, k)\)-insensitive graph \(G\), where \(k = 2\) and \(n \geq 3\gamma(G)^2 – 2\gamma(G) + 3\) that holds for \(\gamma(G) \geq 3\). For small \(n\), we present tighter bounds (in some cases exact values) for this minimum number of edges.

A queen on a hexagonal board with hexagonal cells is defined as a piece that moves along three lines, namely along the cells in the same row, up diagonal, or down diagonal. A queen dominates a cell if the cell is in the same line as the queen.
We show that hexagonal boards with \(n \geq 1\) rows and diagonals, where \(n \equiv 3 \pmod{4}\), have only two types of minimum dominating sets. We also determine the irredundance numbers of the boards with \(5\) and \(7\) rows.

A well-spread sequence is an increasing sequence of distinct positive integers whose pairwise sums are distinct. Some properties of these sequences are discussed.

In this note, we consider finite, undirected, and simple graphs. A subset \(D\) of the vertex set of a graph \(G\) is a dominating set if each vertex of \(G\) is either in \(D\) or adjacent to some vertex of \(D\). A dominating set of minimum cardinality is called a minimum dominating set.A vertex \(v\) of a graph \(G\) is called a cut-vertex of G if \(G – v\) has more components than \(G\). A block of a graph is a maximal connected subgraph having no cut-vertex.A block-cactus graph is a graph whose blocks are either complete graphs or cycles, and we speak of a cactus if the complete graphs consist of only one edge.In our main theorem, we shall show that the minimum dominating set problem of an arbitrary graph can be reduced to its blocks. This theorem provides a linear-time algorithm for determining a minimum dominating set in a block-cactus graph, and thus, it can be seen as a supplement to a linear-time algorithm for finding a minimum dominating set in a cactus, presented by S.T. Hedetniemi, R.C. Laskar, and J. Pfaff in 1986.

For a graph facility or multi-facility location problem, each vertex is typically considered to be the location for one customer or one facility. Typically, the number of facilities is predetermined, and one must optimally locate these facilities so as to minimize some function of the distances between customers and facilities (and, perhaps, of the distances among the facilities). For example, p facility locations (such as, for hospitals or fire stations) might be chosen so as to minimize the maximum or the average distance from a customer to the nearest facility. The problem investigated in this paper considers all of the facilities to be distinct, and we seek to minimize the average customer-to-facility distance, primarily for grid graphs.